This project involves collaborative research with NIH scientists, in which the research requires sophisticated mathematical tools or an acquaintance with theoretical methods in the physical sciences. Drs. George Weiss, Marian Bogunya, and Alexander Berezhkovskii (NIDDK) have worked on a variety of problems required to develop the theory of single-molecule spectroscopy. So far an exact solution has been found for the simplest possible two-state system which is an isomerization scheme with first-order transitions. Considerable effort has been spent in developing theoretical approaches to deal with more complicated chemical systems. This requires expansion of the theory of cumulative residence times in one or more states of a multistate system. To date we have published several papers on this subject. Drs. George Weiss, Marian Bogunya and Alexander Berezhkovskii (NIDDK) have developed a crude model which couples diffusive motion of molecules, and statistics of the amount of time during which a membrane pore has been occupied by n ions. Some progress has been made on this project, but many quite difficult questions remain.